Characterization of Stationary Gaussian Process

Theorem

Let $S$ be a Gaussian stochastic process giving rise to a time series $T$.

Let the the mean of $S$ be fixed.

Let the autocovariance matrix of $S$ also be fixed.

Then $S$ is stationary.

Proof

From Characterization of Multivariate Gaussian Distribution, the Gaussian distribution is completely characterized by its expectation and its variance.

The result follows.

$\blacksquare$

Sources

• 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):

Part $\text {I}$: Stochastic Models and their Forecasting:

$2$: Autocorrelation Function and Spectrum of Stationary Processes:

$2.1$ Autocorrelation Properties of Stationary Models:

$2.1.3$ Positive Definiteness and the Autocovariance Matrix: Gaussian processes